The sum displaystyle∑10r=1(r2 + 1) × (r!) is equal to :

Question

The sum displaystyle∑10r=1(r2 + 1) × (r!) is equal to : (A) (11 !) (B) 10 × (11!) (C) 101 × (11!) (D) 11 × (11!)

Tardigrade Admin · Accepted Answer

displaystyle∑r=110(r2+1) .r ! = displaystyle∑r=110 (r+1)2-2 r r ! = displaystyle∑r=110(r+1)(r+1) !-2 displaystyle∑r=110 r .r ! = displaystyle∑r=110 (r+1)(r+1) !-r(r !) - displaystyle∑r=110 r .r ! =(11.11 !-1)- displaystyle∑r=110((r+1) !-r !) =(11.11 !-1-(11 !-1 !) = 10.11 !

Q. The sum $r = 1 \sum 10 (r^{2} + 1) \times (r!)$ is equal to :

$(11!)$

$10 \times (11!)$

$101 \times (11!)$

$11 \times (11!)$

$r = 1 \sum 10 (r^{2} + 1) . r!$
$= r = 1 \sum 10 {(r + 1)^{2} - 2 r} r!$
$= r = 1 \sum 10 (r + 1) (r + 1)! - 2 r = 1 \sum 10 r . r!$
$= r = 1 \sum 10 {(r + 1) (r + 1)! - r (r!)} - r = 1 \sum 10 r . r!$
$= (11.11! - 1) - r = 1 \sum 10 ((r + 1)! - r!)$
$= (11.11! - 1 - (11! - 1!)$
$= 10.11!$

Q. The sum r=1∑10​(r2+1)×(r!) is equal to :

(11!)

10×(11!)

101×(11!)

11×(11!)